${\displaystyle {\begin{alignedat}{4}&\log _{10}x;&&&&\int \log _{10}x\,dx&&=0.4343x(\log _{\epsilon }x-1)+C.\\&a^{x}&&{\text{(p.149)}};\quad \quad &&\int a^{x}\,dx&&={\dfrac {a^{x}}{\log _{\epsilon }a}}+C.\\&\cos ax;&&&&\int \cos ax\,dx&&={\frac {1}{a}}\sin ax+C\\\end{alignedat}}}$

${\displaystyle \sin ax;\;\quad \quad \quad \int \sin ax\,dx=-{\frac {1}{a}}\cos ax+C.}$

${\displaystyle \cos 2\theta =\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1}$;

${\displaystyle \cos ^{2}\theta ={\tfrac {1}{2}}(\cos 2\theta +1}$,

${\displaystyle {\begin{aligned}\int \cos ^{2}\theta \,d\theta &={\tfrac {1}{2}}\int (\cos 2\theta +1)\,d\theta \\&={\tfrac {1}{2}}\int \cos 2\theta \,d\theta +{\tfrac {1}{2}}\int d\theta .\\&={\frac {\sin 2\theta }{4}}+{\frac {\theta }{2}}+C.\quad {\text{(See also p. 227.)}}\end{aligned}}}$